Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  kur14lem9 Structured version   Visualization version   GIF version

Theorem kur14lem9 31524
Description: Lemma for kur14 31526. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j 𝐽 ∈ Top
kur14lem.x 𝑋 = 𝐽
kur14lem.k 𝐾 = (cls‘𝐽)
kur14lem.i 𝐼 = (int‘𝐽)
kur14lem.a 𝐴𝑋
kur14lem.b 𝐵 = (𝑋 ∖ (𝐾𝐴))
kur14lem.c 𝐶 = (𝐾‘(𝑋𝐴))
kur14lem.d 𝐷 = (𝐼‘(𝐾𝐴))
kur14lem.t 𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))
kur14lem.s 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}
Assertion
Ref Expression
kur14lem9 (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐾   𝑥,𝑦,𝑇   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑦)

Proof of Theorem kur14lem9
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 kur14lem.s . . 3 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}
2 vex 3343 . . . . . 6 𝑠 ∈ V
32elintrab 4640 . . . . 5 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ↔ ∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥))
4 ssun1 3919 . . . . . . . 8 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)})
5 ssun1 3919 . . . . . . . . 9 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))})
6 ssun1 3919 . . . . . . . . . 10 (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ⊆ ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))
7 kur14lem.t . . . . . . . . . 10 𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))
86, 7sseqtr4i 3779 . . . . . . . . 9 (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ⊆ 𝑇
95, 8sstri 3753 . . . . . . . 8 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ 𝑇
104, 9sstri 3753 . . . . . . 7 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
11 kur14lem.j . . . . . . . . . . 11 𝐽 ∈ Top
12 kur14lem.x . . . . . . . . . . . 12 𝑋 = 𝐽
1312topopn 20933 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝑋𝐽)
1411, 13ax-mp 5 . . . . . . . . . 10 𝑋𝐽
1514elexi 3353 . . . . . . . . 9 𝑋 ∈ V
16 kur14lem.a . . . . . . . . 9 𝐴𝑋
1715, 16ssexi 4955 . . . . . . . 8 𝐴 ∈ V
1817tpid1 4447 . . . . . . 7 𝐴 ∈ {𝐴, (𝑋𝐴), (𝐾𝐴)}
1910, 18sselii 3741 . . . . . 6 𝐴𝑇
20 kur14lem.k . . . . . . . . 9 𝐾 = (cls‘𝐽)
21 kur14lem.i . . . . . . . . 9 𝐼 = (int‘𝐽)
22 kur14lem.b . . . . . . . . 9 𝐵 = (𝑋 ∖ (𝐾𝐴))
23 kur14lem.c . . . . . . . . 9 𝐶 = (𝐾‘(𝑋𝐴))
24 kur14lem.d . . . . . . . . 9 𝐷 = (𝐼‘(𝐾𝐴))
2511, 12, 20, 21, 16, 22, 23, 24, 7kur14lem7 31522 . . . . . . . 8 (𝑦𝑇 → (𝑦𝑋 ∧ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
2625simprd 482 . . . . . . 7 (𝑦𝑇 → {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)
2726rgen 3060 . . . . . 6 𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇
2825simpld 477 . . . . . . . . . 10 (𝑦𝑇𝑦𝑋)
2915elpw2 4977 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3028, 29sylibr 224 . . . . . . . . 9 (𝑦𝑇𝑦 ∈ 𝒫 𝑋)
3130ssriv 3748 . . . . . . . 8 𝑇 ⊆ 𝒫 𝑋
3215pwex 4997 . . . . . . . . 9 𝒫 𝑋 ∈ V
3332elpw2 4977 . . . . . . . 8 (𝑇 ∈ 𝒫 𝒫 𝑋𝑇 ⊆ 𝒫 𝑋)
3431, 33mpbir 221 . . . . . . 7 𝑇 ∈ 𝒫 𝒫 𝑋
35 eleq2 2828 . . . . . . . . . 10 (𝑥 = 𝑇 → (𝐴𝑥𝐴𝑇))
36 sseq2 3768 . . . . . . . . . . 11 (𝑥 = 𝑇 → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3736raleqbi1dv 3285 . . . . . . . . . 10 (𝑥 = 𝑇 → (∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3835, 37anbi12d 749 . . . . . . . . 9 (𝑥 = 𝑇 → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)))
39 eleq2 2828 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑠𝑥𝑠𝑇))
4038, 39imbi12d 333 . . . . . . . 8 (𝑥 = 𝑇 → (((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) ↔ ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇)))
4140rspccv 3446 . . . . . . 7 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → (𝑇 ∈ 𝒫 𝒫 𝑋 → ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇)))
4234, 41mpi 20 . . . . . 6 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇))
4319, 27, 42mp2ani 716 . . . . 5 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → 𝑠𝑇)
443, 43sylbi 207 . . . 4 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} → 𝑠𝑇)
4544ssriv 3748 . . 3 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ⊆ 𝑇
461, 45eqsstri 3776 . 2 𝑆𝑇
4711, 12, 20, 21, 16, 22, 23, 24, 7kur14lem8 31523 . 2 (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ 14)
48 1nn0 11520 . . 3 1 ∈ ℕ0
49 4nn0 11523 . . 3 4 ∈ ℕ0
5048, 49deccl 11724 . 2 14 ∈ ℕ0
5146, 47, 50hashsslei 13425 1 (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  cdif 3712  cun 3713  wss 3715  𝒫 cpw 4302  {cpr 4323  {ctp 4325   cuni 4588   cint 4627   class class class wbr 4804  cfv 6049  Fincfn 8123  1c1 10149  cle 10287  4c4 11284  cdc 11705  chash 13331  Topctop 20920  intcnt 21043  clsccl 21044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-xnn0 11576  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-hash 13332  df-top 20921  df-cld 21045  df-ntr 21046  df-cls 21047
This theorem is referenced by:  kur14lem10  31525
  Copyright terms: Public domain W3C validator